Definition:Conic Section/Focus

This page is about focus of conic section. For other uses, see focus.

Definition

Let $\KK$ be a conic section specified in terms of:

a given straight line $D$

a given point $F$

a given constant $\epsilon$

where $K$ is the locus of points $P$ such that the distance $p$ from $P$ to $D$ and the distance $q$ from $P$ to $F$ are related by the condition:

$q = \epsilon \, p$

The point $F$ is known as the focus of $\KK$.

Also see

• Definition:Focus of Ellipse
• Definition:Focus of Parabola
• Definition:Focus of Hyperbola

• Definition:Directrix of Conic Section
• Definition:Focus-Directrix Property of Conic Section

• Results about foci of conic sections can be found here.

Historical Note

The focus-directrix definition of a conic section was first documented by Pappus of Alexandria.

It appears in his Collection.

As he was scrupulous in documenting his sources, and he gives none for this construction, it can be supposed that it originated with him.

Linguistic Note

The word focus is of Latin origin, hence its irregular plural form foci.

It was introduced into geometry by Johannes Kepler when he established his First Law of Planetary Motion.

The word in Latin means fireplace or hearth, which is appropriate, considering the position of the sun.

The pronunciation of foci has a hard c, and is rendered approximately as foke-eye.

Beware the solecism of pronouncing it fo-sigh, which is incorrect.

Sources

• 1933: D.M.Y. Sommerville: Analytical Conics (3rd ed.) ... (previous) ... (next): Chapter $\text {IV}$. The Ellipse: $1 \text a$. Focal properties
• 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {A}.8$: Pappus (fourth century A.D.): Appendix: The Focus-Directrix-Eccentricity Definitions of the Conic Sections
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): conic (conic section)
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): focus (plural foci)
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): conic (conic section)
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): focus (plural foci)